#10086357

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(2,4)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,26)(27,30)(28,31)(29,34)(32,38)(33,35)(36,40)(37,39);(1,2,5,4)(3,7,8,6)(10,14)(15,21)(16,22)(23,24)(25,26)(27,30)(28,33)(29,32)(31,35)(34,38)(36,40)(37,39);(1,3,5,8)(2,6,4,7)(11,17)(15,21)(18,19)(23,24)(26,27)(28,35)(32,34)(36,37);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27,30,26)(28,33,35,31)(29,34,38,32)(36,40,37,39);(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,28,30,35)(26,31,27,33)(29,37,38,36)(32,40,34,39);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)

Polycyclic presentation:

2^9|b^2=hi,c^2=hi,d^2=i,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9993553);